Theorem dfac5 10128

Description: Equivalence of two versions of the Axiom of Choice. The right-hand side is Theorem 6M(4) of [Enderton] p. 151 and asserts that given a family of mutually disjoint nonempty sets, a set exists containing exactly one member from each set in the family. The proof does not depend on AC. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
dfac5	⊢ (CHOICE ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))

Distinct variable group:   𝑥,𝑧,𝑦,𝑤,𝑣

Proof of Theorem dfac5

Dummy variables 𝑓 ℎ 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac4 10122	. . 3 ⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)))
2		neeq1 3022	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
3	2	cbvralvw 3245	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ↔ ∀𝑤 ∈ 𝑥 𝑤 ≠ ∅)
4	3	anbi2i 635	. . . . . . . . . . 11 ⊢ ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) ↔ (∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑤 ∈ 𝑥 𝑤 ≠ ∅))
5		r19.26 3127	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ 𝑥 ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) ↔ (∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑤 ∈ 𝑥 𝑤 ≠ ∅))
6	4, 5	bitr4i 281	. . . . . . . . . 10 ⊢ ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) ↔ ∀𝑤 ∈ 𝑥 ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅))
7		pm3.35 815	. . . . . . . . . . . 12 ⊢ ((𝑤 ≠ ∅ ∧ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → (𝑓‘𝑤) ∈ 𝑤)
8	7	ancoms 464	. . . . . . . . . . 11 ⊢ (((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → (𝑓‘𝑤) ∈ 𝑤)
9	8	ralimi 3104	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ 𝑥 ((𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑤 ≠ ∅) → ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤)
10	6, 9	sylbi 220	. . . . . . . . 9 ⊢ ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) → ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤)
11		r19.26 3127	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
12		elin 3922	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ ran 𝑓))
13		fvelrnb 6945	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 Fn 𝑥 → (𝑣 ∈ ran 𝑓 ↔ ∃𝑡 ∈ 𝑥 (𝑓‘𝑡) = 𝑣))
14	13	biimpac 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑣 ∈ ran 𝑓 ∧ 𝑓 Fn 𝑥) → ∃𝑡 ∈ 𝑥 (𝑓‘𝑡) = 𝑣)
15		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → (𝑓‘𝑤) = (𝑓‘𝑡))
16		id 23	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → 𝑤 = 𝑡)
17	15, 16	eleq12d 2859	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = 𝑡 → ((𝑓‘𝑤) ∈ 𝑤 ↔ (𝑓‘𝑡) ∈ 𝑡))
18		neeq2 3023	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → (𝑧 ≠ 𝑤 ↔ 𝑧 ≠ 𝑡))
19		ineq2 4167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = 𝑡 → (𝑧 ∩ 𝑤) = (𝑧 ∩ 𝑡))
20	19	eqeq1d 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = 𝑡 → ((𝑧 ∩ 𝑤) = ∅ ↔ (𝑧 ∩ 𝑡) = ∅))
21	18, 20	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = 𝑡 → ((𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) ↔ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)))
22	17, 21	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 = 𝑡 → (((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) ↔ ((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅))))
23	22	rspcv 3579	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ 𝑥 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅))))
24		minel 4426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ∩ 𝑡) = ∅) → ¬ (𝑓‘𝑡) ∈ 𝑧)
25	24	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓‘𝑡) ∈ 𝑡 → ((𝑧 ∩ 𝑡) = ∅ → ¬ (𝑓‘𝑡) ∈ 𝑧))
26	25	imim2d 58	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑡) ∈ 𝑡 → ((𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅) → (𝑧 ≠ 𝑡 → ¬ (𝑓‘𝑡) ∈ 𝑧)))
27	26	imp 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) → (𝑧 ≠ 𝑡 → ¬ (𝑓‘𝑡) ∈ 𝑧))
28	27	necon4ad 2979	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) → ((𝑓‘𝑡) ∈ 𝑧 → 𝑧 = 𝑡))
29		eleq1 2853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓‘𝑡) = 𝑣 → ((𝑓‘𝑡) ∈ 𝑧 ↔ 𝑣 ∈ 𝑧))
30	29	biimpar 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧) → (𝑓‘𝑡) ∈ 𝑧)
31	28, 30	impel 515	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) ∧ ((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧)) → 𝑧 = 𝑡)
32		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = 𝑡 → (𝑓‘𝑧) = (𝑓‘𝑡))
33		eqeq2 2777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑡) = 𝑣 → ((𝑓‘𝑧) = (𝑓‘𝑡) ↔ (𝑓‘𝑧) = 𝑣))
34		eqcom 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑓‘𝑧) = 𝑣 ↔ 𝑣 = (𝑓‘𝑧))
35	33, 34	bitrdi 290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑓‘𝑡) = 𝑣 → ((𝑓‘𝑧) = (𝑓‘𝑡) ↔ 𝑣 = (𝑓‘𝑧)))
36	32, 35	imbitrid 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓‘𝑡) = 𝑣 → (𝑧 = 𝑡 → 𝑣 = (𝑓‘𝑧)))
37	36	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) ∧ ((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧)) → (𝑧 = 𝑡 → 𝑣 = (𝑓‘𝑧)))
38	31, 37	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) ∧ ((𝑓‘𝑡) = 𝑣 ∧ 𝑣 ∈ 𝑧)) → 𝑣 = (𝑓‘𝑧))
39	38	exp32 426	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓‘𝑡) ∈ 𝑡 ∧ (𝑧 ≠ 𝑡 → (𝑧 ∩ 𝑡) = ∅)) → ((𝑓‘𝑡) = 𝑣 → (𝑣 ∈ 𝑧 → 𝑣 = (𝑓‘𝑧))))
40	23, 39	syl6com 38	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑡 ∈ 𝑥 → ((𝑓‘𝑡) = 𝑣 → (𝑣 ∈ 𝑧 → 𝑣 = (𝑓‘𝑧)))))
41	40	com14 97	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 ∈ 𝑧 → (𝑡 ∈ 𝑥 → ((𝑓‘𝑡) = 𝑣 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧)))))
42	41	rexlimdv 3166	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ 𝑧 → (∃𝑡 ∈ 𝑥 (𝑓‘𝑡) = 𝑣 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧))))
43	14, 42	syl5 35	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ 𝑧 → ((𝑣 ∈ ran 𝑓 ∧ 𝑓 Fn 𝑥) → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧))))
44	43	expd 421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ 𝑧 → (𝑣 ∈ ran 𝑓 → (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → 𝑣 = (𝑓‘𝑧)))))
45	44	com4t 94	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ 𝑧 → (𝑣 ∈ ran 𝑓 → 𝑣 = (𝑓‘𝑧)))))
46	45	imp4b 427	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
47	12, 46	biimtrid 245	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 ((𝑓‘𝑤) ∈ 𝑤 ∧ (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
48	11, 47	sylan2br 607	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 Fn 𝑥 ∧ (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
49	48	anassrs 473	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
50	49	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) → 𝑣 = (𝑓‘𝑧)))
51		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑧 → (𝑓‘𝑤) = (𝑓‘𝑧))
52		id 23	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑧 → 𝑤 = 𝑧)
53	51, 52	eleq12d 2859	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑧 → ((𝑓‘𝑤) ∈ 𝑤 ↔ (𝑓‘𝑧) ∈ 𝑧))
54	53	rspcv 3579	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (𝑓‘𝑧) ∈ 𝑧))
55		fnfvelrn 7079	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 Fn 𝑥 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ ran 𝑓)
56	55	expcom 419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ 𝑥 → (𝑓 Fn 𝑥 → (𝑓‘𝑧) ∈ ran 𝑓))
57	54, 56	anim12d 621	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑥 → ((∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ 𝑓 Fn 𝑥) → ((𝑓‘𝑧) ∈ 𝑧 ∧ (𝑓‘𝑧) ∈ ran 𝑓)))
58		elin 3922	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓) ↔ ((𝑓‘𝑧) ∈ 𝑧 ∧ (𝑓‘𝑧) ∈ ran 𝑓))
59	57, 58	imbitrrdi 255	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑥 → ((∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 ∧ 𝑓 Fn 𝑥) → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓)))
60	59	expd 421	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (𝑓 Fn 𝑥 → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓))))
61	60	com13 89	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓))))
62	61	imp31 423	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓))
63		eleq1 2853	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑓‘𝑧) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ (𝑓‘𝑧) ∈ (𝑧 ∩ ran 𝑓)))
64	62, 63	syl5ibrcom 250	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (𝑣 = (𝑓‘𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
65	64	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 = (𝑓‘𝑧) → 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
66	50, 65	impbid 215	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧)))
67	66	ex 418	. . . . . . . . . . . . 13 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
68	67	alrimdv 1962	. . . . . . . . . . . 12 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
69		fvex 6898	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑧) ∈ V
70		eqeq2 2777	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑓‘𝑧) → (𝑣 = ℎ ↔ 𝑣 = (𝑓‘𝑧)))
71	70	bibi2d 345	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓‘𝑧) → ((𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ) ↔ (𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
72	71	albidv 1953	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓‘𝑧) → (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ) ↔ ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧))))
73	69, 72	spcev 3567	. . . . . . . . . . . . 13 ⊢ (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧)) → ∃ℎ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ))
74		eu6 2604	. . . . . . . . . . . . 13 ⊢ (∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ ∃ℎ∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = ℎ))
75	73, 74	sylibr 237	. . . . . . . . . . . 12 ⊢ (∀𝑣(𝑣 ∈ (𝑧 ∩ ran 𝑓) ↔ 𝑣 = (𝑓‘𝑧)) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))
76	68, 75	syl6 36	. . . . . . . . . . 11 ⊢ (((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) ∧ 𝑧 ∈ 𝑥) → (∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
77	76	ralimdva 3179	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤) → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
78	77	ex 418	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 (𝑓‘𝑤) ∈ 𝑤 → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
79	10, 78	syl5 35	. . . . . . . 8 ⊢ (𝑓 Fn 𝑥 → ((∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) ∧ ∀𝑧 ∈ 𝑥 𝑧 ≠ ∅) → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓))))
80	79	expd 421	. . . . . . 7 ⊢ (𝑓 Fn 𝑥 → (∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) → (∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ → (∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))))
81	80	imp4b 427	. . . . . 6 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
82		vex 3461	. . . . . . . 8 ⊢ 𝑓 ∈ V
83	82	rnex 7913	. . . . . . 7 ⊢ ran 𝑓 ∈ V
84		ineq2 4167	. . . . . . . . . 10 ⊢ (𝑦 = ran 𝑓 → (𝑧 ∩ 𝑦) = (𝑧 ∩ ran 𝑓))
85	84	eleq2d 2851	. . . . . . . . 9 ⊢ (𝑦 = ran 𝑓 → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
86	85	eubidv 2616	. . . . . . . 8 ⊢ (𝑦 = ran 𝑓 → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
87	86	ralbidv 3190	. . . . . . 7 ⊢ (𝑦 = ran 𝑓 → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓)))
88	83, 87	spcev 3567	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ ran 𝑓) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
89	81, 88	syl6 36	. . . . 5 ⊢ ((𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
90	89	exlimiv 1963	. . . 4 ⊢ (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
91	90	alimi 1844	. . 3 ⊢ (∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑤 ∈ 𝑥 (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)) → ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
92	1, 91	sylbi 220	. 2 ⊢ (CHOICE → ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
93		eqid 2765	. . . . 5 ⊢ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))}
94		biid 264	. . . . 5 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
95		eqid 2765	. . . . 5 ⊢ (∪ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦) = (∪ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))} ∩ 𝑦)
96	93, 94, 95	dfac5lem5 10127	. . . 4 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
97	96	alrimiv 1960	. . 3 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∀ℎ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
98		dfac3 10121	. . 3 ⊢ (CHOICE ↔ ∀ℎ∃𝑓∀𝑤 ∈ ℎ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
99	97, 98	sylibr 237	. 2 ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → CHOICE)
100	92, 99	impbii 212	1 ⊢ (CHOICE ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃!weu 2598  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ∩ cin 3905  ∅c0 4286  {csn 4591  ∪ cuni 4874   × cxp 5661  ran crn 5664   Fn wfn 6535  ‘cfv 6540  CHOICEwac 10115

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ac 10116

This theorem is used by:  dfackm  10166  ac8  10491  dfac5prim  45759